Practice Maths

Rational and Irrational Numbers

Key Ideas

Key Terms

rational number
Can be written as a fraction p/q where p and q are integers and q ≠ 0.
irrational number
Cannot be written as a fraction of two integers. Its decimal expansion is non-terminating and non-recurring.
real numbers
(ℝ) includes all rational and irrational numbers — every point on the number line.
ConceptRule / Example
RationalCan write as p/q — e.g. 3/4, −7, 0.625, 0.333…
IrrationalCannot write as p/q — e.g. √2, √5, π
Perfect square test√9 = 3 (rational); √10 is irrational (10 not a perfect square)
Recurring → fractionx = 0.777… ⇒ 10x = 7.777… ⇒ 9x = 7 ⇒ x = 7/9
Real numbersℝ = ℚ ∪ Irrationals — every point on the number line

The Real Number Hierarchy

Real Numbers (ℝ) contain Rational Numbers (ℚ) and Irrational Numbers. Within ℚ: Integers (ℤ) ⊃ Natural Numbers (ℕ).

Hot Tip A square root is rational only when the number under the root is a perfect square. √9 = 3 (rational), but √10 is irrational because 10 is not a perfect square.

Worked Example

Question: Classify each number as rational or irrational:   0.333…,   √16,   √7,   22/7

0.333… — Recurring decimal = 1/3. Rational.

√16 — √16 = 4 = 4/1. Rational.

√7 — 7 is not a perfect square; 2.6457… is non-terminating, non-recurring. Irrational.

22/7 — Already in form p/q. (It approximates π but is itself rational.) Rational.

Worked Example — Converting a Recurring Decimal to a Fraction

Question: Convert 0.777… to a fraction.

Step 1: Let x = 0.777…

Step 2: Multiply: 10x = 7.777…

Step 3: Subtract: 9x = 7

Step 4: Solve: x = 7/9

What Are Rational Numbers?

A rational number is any number that can be written in the form p/q, where p and q are both integers and q ≠ 0. The word "rational" comes from "ratio" — it literally means a number that can be expressed as a ratio of two whole numbers.

Examples of rational numbers include: 3/4, −5/2, 7 (which equals 7/1), 0.6 (which equals 3/5), and 0.333... (which equals 1/3). Notice that whole numbers and fractions are all rational numbers.

The decimal expansion of a rational number either terminates (ends, like 0.75) or recurs (repeats in a pattern, like 0.363636... = 4/11). If a decimal does either of these things, it is rational.

What Are Irrational Numbers?

An irrational number cannot be written as p/q for any integers p and q. Its decimal expansion goes on forever with no repeating pattern. You can never write it exactly as a fraction or a terminating/recurring decimal.

The most famous examples are: √2 ≈ 1.41421356..., √3 ≈ 1.73205080..., and π ≈ 3.14159265.... The digits never end and never settle into a repeating cycle.

A useful rule: √n is irrational whenever n is not a perfect square. So √4 = 2 is rational, but √5, √6, √7 are all irrational.

The Real Number System

Together, rational and irrational numbers form the real numbers. You can think of this as a hierarchy:

Real Numbers contain everything. Inside real numbers, you have two separate groups that do not overlap: Rational Numbers and Irrational Numbers. Inside rational numbers, you have Integers (..., −2, −1, 0, 1, 2, ...), and inside integers you have Natural Numbers (1, 2, 3, ...). Every natural number is an integer, every integer is rational, every rational number is real — but not every real number is rational.

How to Classify a Number

To decide whether a number is rational or irrational, ask yourself three questions in order. First: is it a whole number or a simple fraction? If yes, rational. Second: is it the square root of a perfect square? (Like √9 = 3 or √25 = 5?) If yes, rational. Third: does its decimal expansion terminate or repeat? If yes, rational. If none of these apply, it is irrational.

For example: √16 = 4 (rational, it equals a whole number). √17 is irrational (17 is not a perfect square). 0.125 = 1/8 (rational, it terminates). 0.101001000100001... (irrational, it never repeats).

Common exam mistake: Students sometimes think that because π appears in a formula like the area of a circle (A = πr2), the answer must always be irrational. But if r = 0 the area is 0 (rational!), and if the question says "give your answer in terms of π" you are leaving the irrational part as a symbol. Always check whether the π cancels or not before classifying your answer.

Mastery Practice

  1. Classify each number as Rational or Irrational. Complete the table. Fluency

      Number Rational or Irrational?
    (a)√4 
    (b)√2 
    (c)π 
    (d)0.333… 
    (e)1.5 
    (f)−7 
    (g)0 
    (h)√9 
    (i)√5 
    (j)5/8 

  2. Order each set of numbers from smallest to largest. Find approximate decimal values to help. Fluency

      Numbers to order Smallest → Largest
    (a)√5,   2.1,   √3,   7/3,   π−1 
    (b)−1/2,   −√2,   −0.7,   −1 
    (c)π,   22/7,   3.14,   √10,   3.2 

  3. Convert each recurring decimal to a fraction in simplest form. Fluency

      Recurring Decimal Fraction
    (a)0.5555… 
    (b)0.2222… 
    (c)0.9999… 
    (d)0.181818… 
    (e)0.363636… 
    (f)0.1666… 
    (g)0.8333… 
    (h)0.142857142857… 

  4. State whether each claim is True or False, and give a brief justification. Fluency

    1. Every integer is a rational number.
    2. Every rational number is an integer.
    3. √25 is irrational because it contains a square root sign.
    4. 0.999… is less than 1.
    5. The sum of two irrational numbers is always irrational.
    6. 22/7 is the exact value of π.

  5. For each number, identify all number sets it belongs to from: Natural (ℕ), Integer (ℤ), Rational (ℚ), Irrational, Real (ℝ). Understanding

    Number Hierarchy. Recall that ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ, and that irrational numbers are in ℝ but not ℚ.
    1. 12
    2. −5
    3. 3/4
    4. √2
    5. 0
    6. −2.7
    7. √81
    8. π/2

  6. Square garden problem. Understanding

    Garden Design. A square garden has an area of 50 m². A gardener claims the side length is “about 7 metres.”
    1. Find the exact side length, leaving your answer in surd form.
    2. Is the side length rational or irrational? Explain.
    3. Use a calculator to verify whether 7 m is a reasonable approximation.

  7. Circumference of a circular path. Understanding

    Physics Lab. A physicist measures the circumference of a circular particle path as 5π metres and writes “the circumference is exactly 15.708 metres.”
    1. Identify the error in the physicist’s statement.
    2. Is 5π rational or irrational? Explain.
    3. Write a more accurate statement about the circumference.

  8. Analyse each student claim about rational and irrational numbers. Understanding

    Student Debate. Three students make claims about real numbers. Decide if each claim is correct and explain your reasoning with examples.
    1. Amir says: “Every decimal that goes on forever must be irrational.” Is he correct?
    2. Bella says: “The product of two irrational numbers is always irrational.” Test this with √2 × √2, √3 × √3, and √2 × √8. What is your conclusion?
    3. Carlos says: “Between any two rational numbers there is always another rational number.” Give an example to illustrate this.

  9. Proving that √2 is irrational. Problem Solving

    Proof by Contradiction. Suppose √2 = p/q in simplest form, so p and q share no common factors. Use algebra to derive a contradiction.
    1. Square both sides of √2 = p/q to show that p² = 2q².
    2. Explain why p² must be even, and therefore p must be even.
    3. Write p = 2k and substitute to show q must also be even.
    4. Explain the contradiction and conclude that √2 is irrational.

  10. The Golden Ratio — an irrational number in nature. Problem Solving

    The Golden Ratio. The golden ratio is defined as φ = (1 + √5)/2 ≈ 1.618… It appears in art, architecture, and nature.
    1. Is φ rational or irrational? Give a clear reason.
    2. Calculate φ² using the surd form and show that φ² = φ + 1.
    3. Use the recurring decimal method to explain why 0.9999… = 1 exactly. What does this tell you about decimal representations?
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